Sunday, April 24, 2005

The world of classical worlds is zero-dimensional

Also this day meant a considerable progress in the understanding of quantum TGD in terms of hyper-finite type II_1 factors. An interesting question is how much besides II_1 factors, their inclusions, and generalized Feynman diagrammatics is needed to end up with TGD. In particular, do the crucial number theoretical ideas follow from the framework of von Neumann algebras? The world of classical worlds is zero-dimensional Configuration space CH corresponds to the space of 3-surfaces of 8-D imbedding sapce, the world of classical worlds. Its tangent space corresponds to the sub-space of gamma matrices of CH Clifford algebra having von Neumann dimension 1 and thus has dimension D= 2log_2(Tr(Id)=1)=0 obtained by generalizing the formula applying in finite-dimensions. A rather paradoxical looking result indeed but understandable since the trace of the projector to this sub-space is infinitely smaller than the trace of identity which equals to one. The generalization of the notion of number by allowing infinite primes, integers and rationals however led already earlier to the result that there is infinite variety of numbers which differ from each other by multiplication with numbers which are units in the real sense but have varying p-adic norms. Single space-time point becomes infinitely structured in the sense of number theory. What more could a number mystic dream than algebraic holography: single space-time point containing in its structure the configuration space of all classical worlds! Loop becomes closed: from point to infinite-dimensional space of classical worlds which turns out to be the point! Brahman=Atman taken to extreme! Configuration space tangent space as logarithm of quantum plane The identification of the tangent space of configuration space as a subspace of gamma matrices allows a natural imbeddding to the Clifford algebra and can be regarded as log_2(M:N)<=2-dimensional module for various sub-factors. One can say that the quantum dimension of configuration space as N module is never larger than 2. Note that the dimension of configuration space as N module occurs in the proposed formula for hbar. State space has quantum dimension D<=8 Bosonic and fermionic sectors of the state space correspond both to II_1 sectors by super-symmetry: thus the dimension is D<=4 for both quark and lepton sectors (it is essential that couplings to the Kähler form of CP_2 are different) and the total quantum dimension is d=4log_2(M:N)<=8 as N module. Hence the quantum counterpart of the imbedding space seems to be in question. Quantum version D=8 Bott periodicity probably holds true so that by self-referential property of von Neumann algebras imbedding space dimension is unique. How effective 2-dimensionality can be consistent with 4-dimensionality The understanding about how effective 2-dimensionality can be consistent with space-time dimension D=4 improved also considerably. Classical non-determinism of Kähler action and cognitive interpretation are decisive in this respect. At the basic level the physics is 2-dimensional but the classical non-determinism making possible cognitive states means that additional dimensions emerge as two parameters appearing in the direct integrals of II_1 factors. These parameters correspond to the two light-like Hamilton Jacobi coordinates labelling partonic 2-surfaces appearing in all known classical solutions of field equations. There is somewhat different manner to state it: same 3-surface X^3 can corresponds to a large number of space-time surfaces X^4(X^3) by classical non-determinism and the "position" of partonic 2-surface characterizes this non-determinism and defines the third spatial coordinate as dynamical coordinate. One further manner to see the emergence of time. Classical non-determinism makes possible time-like entanglement and cognitive states. Without classical non-determinism TGD would reduce to a string model. If string models were correct, our Universe would not be able to form self representations and there would be no string theorists. Thus the mere existence of string theorists proves that they are wrong;-)! For more details see the chapter Was von Neumann Right After All? of TGD. Matti Pitkanen

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